A slight amount of thought will suffice to show that in this elaborate array of argumentation, Descartes is the victim of a subtle fallacy. Our conception of necessary existence cannot include the fact of necessary existence, for (to repeat what we have already said) the one is an ideal concept of the mind, the other is a fact of real existence. The one demands an object beyond the mind conceiving it, the other does not. All that the Cartesian argument could prove would be that the mental concept was necessary, not that the concept had a counterpart in the outer universe. It is, indeed, a necessary judgment that the three angles of a triangle are equal to two right angles, because this is _an identical proposition_; the subject and the predicate are the same, the one being only an expansion of the other. We cannot, therefore, destroy the predicate and leave the subject intact. But it is otherwise when we affirm that any triangular object _exists_, we may then destroy the predicate 'existence,' and yet leave the subject (the notion of the triangle) intact in the mind. It is true that Descartes has not limited himself to this futile _à priori_ demonstration. He has buttressed his formal ontology by a much more suggestive though logically as inconclusive an argument. He again reasons thus in his 'Principles:' We have the idea of an all-perfect being in the mind, but whence do we derive it? It is impossible that we can have an idea of anything, unless there be an original somewhere in the universe whence we derive it, as the shadow is the sign of a substance that casts it. But it is manifest that the more perfect cannot arise from the less perfect, and that which knows something more perfect than itself is not the cause of its own being. Since, therefore, we ourselves are not so perfect as the idea of perfection which we find within us, we are forced to believe that this idea in us is derived from a more perfect being above us, and consequently that such a being exists. It will be observed that this second argument of Descartes is partly cosmological,--though ultimately it merges in the ontological, and falls back upon it for support. Hence, Descartes himself called it an _à posteriori_ argument. And it may therefore serve as a link of connection and transition to the second class of arguments.